Description
Generically in quantum machanics and quantum field theories (and actually even in classical models), whenever we do not know how to compute some physical quantity exactly, we might try to set up a perturbative expansion. If we have a small parameter x we can try to perform our computation for x=0 and then assume that, when we turn on x, we will generate a small correction perturbing the initial calculation, say of order O(x). Then we can try to compute higher and higher order corrections to the unperturbed, i.e. x=0, calculation, in the hope that all these corrections will be less and less important. This process goes under the name of A famous example of this procedure is the calculation of the electron gyromagnetic ratio, g-2, in quantum electrodynamics. This physical quantity has been computed in perturbation theory up to the fourth order in the small parameter expansion (the electroweak coupling constant) matching the experimental value to more than 10 significant digits. We might then wonder if, by including all the perturbative corrections, i.e. by resumming a power series, we can obtain the exact quantity we wanted to study as a function of our small parameter x. In many relevant physical models it turns out that this is not actually possible in any straightforward manner. The corrections coming from higher and higher orders in perturbation theory after decreasing for the first few orders, will start growing extremely fast (factorially to be precise) with the order of our calculation, thus giving a vanishing radius of convergence for the power series. The aim of this project is to understand how to actually makes sense of these asymptotic series and understand why this phenomenon happens very generically in quantum mechanics and QFTs. Group ProjectThis group project will explore the role of asymptotic series and Borel summability in quantum mechanics, with a focus on how divergent perturbation expansions can still yield physically meaningful results. The project will involve reviewing key theoretical concepts, analysing illustrative examples such as the anharmonic oscillator, and presenting both the mathematical framework and its physical interpretation. The use of symbolic programming language such as Mathematica, or Maple is strongly advised. By the end of the group project: Mode of operation and evidence of learning for the group projectThe group will operate through regular meetings, task allocation, and collaborative discussion of key concepts, potentially incorporating the use of Mathematica and Maple for symbolic computation, visualisation, and exploration of examples. Evidence of learning will be demonstrated through comparing notes, supported by computational outputs, reflecting understanding of asymptotic series and Borel summability, as well as the ability to apply and communicate these ideas clearly. Individual ProjectThis individual project will investigate the role of asymptotic series and Borel summability in quantum mechanics, with an emphasis on understanding how divergent perturbative expansions can encode meaningful physical information. The work will include a study of standard examples, such as the anharmonic oscillator, alongside a careful examination of the mathematical structure of asymptotic expansions and their resummation via Borel techniques. Potential advanced topics may include: The project will involve independent reading of research-level material, critical analysis of key results, and the development of clear explanations of both the mathematical framework and its physical implications. Mode of operation and evidence of learning for the individual projectThe project will be conducted through independent study, structured reading, and regular progress reviews to refine understanding and address challenges. Evidence of learning will be demonstrated through weekly worked examples and derivations that show clear comprehension of asymptotic methods and Borel summability as well as a written report. Pre-requisitesCo-requisitesResources |